Abstract
Finite field Fourier transforms are of great interest in coding and cryptography. They are, in particular, used for describing BCH and RS codes in the spectral domain and for representing the solutions of recurrence equations used in stream ciphers. So far, finite field Fourier transforms have only been defined on vectors that have a length which is relatively prime to the characteristic of the field. The aim of the paper is to generalize this definition to arbitrary lengths. Many properties get a simpler interpretation with this approach.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R.E. Blahut Theory and Practice of Error Control Codes Addison-Wesley Publishing Company, Inc., 1983.
R.E. Blahut, “Transform techniques for error control,” IBM J. Res. and Dey. Vol. 23, pp. 299–315, May 1979.
P. Mathys, “A generalization of the discrete Fourier transform in finite fields,” Proc. IEEE Symp. Inform. Theory San Diego (CA), Jan. 14–19, 1990.
S.R. Blackburn, “A Generalization of the Discrete Fourier Transform: Determining the Minimal Polynomial of a Periodic Sequence,” IEEE Trans. Inform. Theory (to appear).
C.G. Günther, “Fourier transform in cryptography and coding,” IEEE Workshop on Inform. Theory Bellagio, Italy, June 1987.
E. Lucas, “Théorie des fonctions numérique simplement périodiques” American Journal of Mathematics vol. 1, pp. 184–321, 1878
H.F. Mattson and G. Solomon, “A new treatment of Bose Chaudhuri codes,” J. Soc. Indus. Appl. Math. Vol. 9, pp. 654–659, 1961.
H. Hasse, “Theorie der höheren Differentiale in einem algebraischen Funktionenkörper mit vollkommenem Konstantenkörper bei beliebiger Charakteristik,” J. reine angew. Math. Bd. 175, S. 50–54, 1936.
G. Castagnoli, J.L. Massey, P.A. Schoeller, and N. Seemann, “On repeated-root cyclic codes,” IEEE Trans. on Inform. Theory vol. IT-37, pp. 337–342, 1991.
L.M. Milne-Thomson The Calculus of Finite Differences MacMillan and Co., London, 1951.
E.L. Key, “An analysis of the structure and complexity of nonlinear sequence generators,” IEEE Trans. on Inform. Theory vol. IT-22, pp. 732–736, 1976.
R.A. Rueppel Analysis and Design of Stream Ciphers Springer-Verlag, Berlin, Heidelberg, 1986.
W.W. Peterson and E.J. Weldon, Jr. Error Correcting Codes MIT Press, Cambridge, MA, and London, 1972.
J.L. Massey, D.J. Costello Jr. and J. Justesen, “Polynomial weights and code constructions,” IEEE Trans. on Inform. Theory vol. IT-19, pp. 101–110, 1973.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media New York
About this chapter
Cite this chapter
Günther, C.G. (1994). A Finite Field Fourier Transform for Vectors of Arbitrary Length. In: Blahut, R.E., Costello, D.J., Maurer, U., Mittelholzer, T. (eds) Communications and Cryptography. The Springer International Series in Engineering and Computer Science, vol 276. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2694-0_15
Download citation
DOI: https://doi.org/10.1007/978-1-4615-2694-0_15
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-6159-6
Online ISBN: 978-1-4615-2694-0
eBook Packages: Springer Book Archive